The purpose of this study is to evaluate a treatment consisting of chemotherapy and surgical removal of cancer followed by additional chemotherapy and then two treatment cycles of very high-dose therapy with reinfusion of patient's own bone marrow derived cells. 2) Overview Inflammatory breast cancer, when treated conventionally, carries a high likelihood of relapse. High doses of combination chemotherapy may improve this outcome for patients and a new combination of drugs and doses is being investigated. To reduce the potential side effects associated with high dose chemotherapy, stem cell rescue (a process in which the patient's own bone marrow derived cells are reinfused back to them) will follow. 3) Who Is Potentially Eligible?: Age approximately 18-65 years; Diagnosed with stage IIIB breast cancer; Must have at least one of the following clinical features within six months from time of diagnosis: inflammation, erythema,pain or hypersensitivity, edema (peau d'orange), thickening of skin. Patient must be in otherwise generally good health. 4) Description of Treatment: If the patient has received no more than one prior cycle chemotherapy treatment and has not had surgical removal of their tumor, they will begin chemotherapy with a drug called doxorubicin. Then, the patient will receive a second drug, Taxol, followed by surgical removal of their breast cancer. All patients will receive Taxol through an intravenous catheter over a period of 96 hours. Granulocyte colony stimulating factor (G-CSF) will be administered to increase the number of circulating bone marrow cells in their blood circulation. Next, the patient will undergo PBSC (peripheral blood stem cell) collection. The procedure involves circulating blood through a machine that separates the blood into components and returns everything but the white blood cells back to the patient.The patient will be admitted to the hospital to receive their first cycle of high-dose combination chemotherapy, consisting of intravenous infusions of doxorubicin and cyclophosphamide, and the next day, an infusion of Taxol. Following this, the patient will receive partial reinfusions of their previously stored PBSC. After no more then a 7-week period, the patient will be readmitted to receive a second cycle of high-dose chemotherapy, consisting of the two drugs melphalan and cisplatin. One week from the beginning of their therapy the patient will receive another dose of melphalan and cisplatin, and following that, partial reinfusions of previously collected PBSC. The patient will receive radiation therapy to the chest wall and lymph node areas following recovery from high-dose chemotherapy, and if the tumor was hormone-receptor positive, will also start taking a drug, tamoxifen, for a period of 5 years. Following treatment, the patient will be closely monitored.